LANDAU-LIFSHITZ MODIFIED FLUID
10.3 Vortex Filaments Motion
From Section10.1we learn that the dynamics generated from ISF converges to those generated from the Euler equation in regular vortices when}→ 0.
However } is never set to zero in an actual simulation. In this section we investigate the contribution of the Landau-Lifshitz term to the motion of a vortex filament.
Recall (Section 9.5) that under a (Berger-)ISF, at each regular point of s, η and s satisfies
∂t∂η+Luˆη = −dp,˜ d? η =0,
∂t∂s+Luˆs= 0
where the perturbed velocity is given by ˆ
uB u−2}2ds−1(s×∆s). (10.4) The value of ˆu at a point on the vortex core reveals the velocity of the moving filament.
In the following calculation we take M as a region in R3, and we assume that the vortex filament has a similar structure as in Fig.10.1. In particular, shas a constant value away from the vortex tube, which has a radius much smaller than the curvature radius of the vortex core. On each orthogonal cross section of the vortex tube, s is a conformal map covering the entire S2once.
Construction of s
Such a functions can be written explicitly in terms of a centroid curve and a stereographic projection from the normal plane of the curve to S2. Let Γ ⊂ M be a space curve arclength-parameterized by γ: I ⊂ R → Γ ⊂ R3, with the Frenet frame (T,N,B), the curvature κ and the torsion τ defined through
T = γ0, T0= κN, B =T×N, N0=−κT +τB.
LetU be a tubular neighborhood of Γ U B
(x ∈ M
dist(x,Γ) < R)
, R 1/κ, R 1/τ. Define the closest pointprojection cp : U → Γ
cp(x) B argmin
γ(`)∈Γ dist x,γ(`)
which makes U −−cp→ Γ a disk bundle. Using the closest point projection, extend the frame (T,N,B) to an orthonormal basis defined onU:
T BT◦cp, N B N◦cp, B B B◦cp .
There are also unique functions r: U → R and θ: U → R/(2π) such that each x ∈U is expressed by
x= cp(x)+r(x)cos(θ(x))N+r(x)sin(θ(x))B.
Now, let F: C→S2be the stereographic projection F: z ∈C7→
2arctan|z|1, arg(z)
∈S2
where the right-hand side is written in spherical coordinates. By function composition, we construct s: U →S2 as
s = F◦Z
where Z: U → Cis an identification of each normal plane cp−1(γ(`)) with the complex plane:
Z(x) B r(r1x)ei(θ(x)+(ϕ◦cp)(x)−(α◦cp)(x)). (10.5) Herer1is a constant withr1 < Rparameterizing the thickness of the vortex filament. In particular the set {|ψ1|2 = |ψ2|2} visualized in Fig. 10.1 agrees with the set {r =r1}. The angle-valued function ϕ: Γ→ R/(2π) is given by ϕ=R `
τ, which is angle between N and a parallel normal vector field ofΓ. The angle-valued functionα: Γ→ R/(2π) is the angle the real axis Z−1(1) made with a parallel normal vector field. In other words,α is a parameter such that α0describes the amount oftwist of vortex lines.
Properties of Stereographic Projection
To compute the perturbed velocity (10.4) we need to evaluate∆s, rotate90 degree, and take the inverse map ds−1. For a first step we compute the Laplacian ∆s. By the chain rule for Laplacian, we have
∆s= ∆(F◦Z)
= X
X∈{T,N,B}
(HessF)Z(dZ(X), dZ(X))+(dF)Z(∆Z).
Therefore, s×∆s= X
X∈{T,N,B}
F Z
×(HessF)Z(dZ(X), dZ(X))+F Z
×(dF)Z(∆Z). (10.6)
This expression can be largely simplified using the following properties of the stereographic projectionF.
First of all, F isconformal, which implies that F
Z
× dF(∆Z) = dF(i∆Z). (10.7) Another fact is that F: C → S2 is harmonic. That is, treating F as an R3- valued function, one has
∆F = −|dF|2F.
In particular, after projecting the result onto the tangent space of S2,
∆S2F =0, F×∆F = 0.
By writing the Laplacian in terms of the Hessian, one also has F
Z ×
(HessF)Z(ζ,ζ)+(HessF)Z(iζ,iζ)
=0 (10.8) for any ζ ∈C.
Later, we will also need to evaluate F Z
×(HessF)Z(i Z,i Z). This quantity is the2nd derivative of F with respect to the angle of the polar coordinate onC.
Lemma10.1. F
Z ×(HessF)Z(i Z,i Z)= (dF)Z
1−|Z|2 1+|Z|2i Z
.
Proof. With the spherical coordinate(A,B) 7→ (cosA, sinAcosB, sinAsinB) the stereographic projection F(Z) is represented by A(Z) = 2arctan|Z1| and B(Z) = arg(Z). In particular, the variation i Z ∈ TZC is mapped to (dF)Z(i Z) = ∂B∂ F. Hence
(dF)Z(i Z) = ∂B∂ F = (0,−sinAsinB, sinAcosB),
(HessF)Z(i Z,i Z) = ∂B∂22F = (0,−sinAcosB,−sinAsinB).
Therefore F
Z
×(HessF)Z(i Z,i Z) = (cosA, sinAcosB, sinAsinB)
×(0,−sinAcosB,−sinAsinB)
= (0, cosAsinAsinB,−cosAsinAcosB)
=−cosA(dF)Z (i Z).
The proof follows after one applies the identity cos(2arctan|Z1|) = |Z|Z||22−+11. Tubular Neighborhood Coordinate
As later we will compute the derivatives of Z (10.5) in (10.6) that is written in the coordinate functions cp, r and θ, we give some useful formulas involving these coordinate functions.
Lemma10.2. The derivatives ofcp,r andθ are dcp= 1−κr1cos(θ)TT[,
dr = cos(θ)N[+sin(θ)B[,
dθ= −1−κrτcos(θ)T[− 1r sin(θ)N[+ 1r cos(θ)B[.
Proof. These derivatives are obtained by taking directional derivatives of cp, r, and θ in the directions T,N,B. Since cp projects orthogonally to Γ, one has dcp(N) = dcp(B) = 0. Now, as one moves infinitesimally along the curve, the normal plane γ(`)+RN(`) +RB(`) turns about a hinge, which is a line parallel to Band passing through the center of the osculat- ing circle. In particular, as ` moves forward by ∆`, a point on the normal plane is lifted by (1−κrcos(θ))∆` in theT direction, depending on the rel- ative position (r,θ) on the normal plane. Therefore, dcp(T)= 1−κr1cos(θ)T. The derivative dr, and the N[,B[ components of dθ follow from the stan- dard cylindrical coordinate. For dθ(T), note that the coordinate vector N turns positively in the rate of τ with respect to the arclength on Γ. Hence the value of θ varies negatively in the rate of τ with respect to arclength.
Thus by chain rule dθ(T) =−τhT, dcp(T)i=−1−κrτcos(θ). Lemma10.3. The covector fieldsT[, N[, B[ satisfy
?d?T[ = 0, ?d?N[ = − κ
1−κrcos(θ), ?d?B[ =0.
Proof. Each of these codifferential quantities is the divergence of the asso- ciated vector field. At the level of vectors, we compute the derivatives (in the following the connection∇R3 on R3 is just the usual d)
∇R3T = d
T◦cp
= ∂
∂`T
hT, dcpi= 1−κrκcos(θ)NT[. Similarly
∇R3N =−1−κrκcos(θ)TT[+1−κrτcos(θ)BT[,
∇R3B =−1−κcos(θ)τ NT[. As a result, the divergence
divT = tr
∇R3T
= X
X∈{T,N,B}
DX,∇R3T(X)E
= 0, divN = X
X∈{T,N,B}
DX,∇R3N(X)E
=−1−κrκcos(θ), divB= X
X∈{T,N,B}
DX,∇R3B(X)E
= 0.
Lemma10.4. The differential of Z defined in(10.5)is given by dZ =
−i1−κrαcos(θ)0 T[+ 1re−iθN[+1rie−iθB[ Z.
Proof. With the aid of Lemma10.2, dZ = r11 dr ei(θ+ϕ◦cp−α◦cp)+rr1i
dθ+(ϕ0−α0)hT, dcpi
ei(θ+ϕ◦cp−α◦cp)
= 1
r
cos(θ)N[+sin(θ)B[ +i
−1−κrαcos(θ)0 T[− 1r sin(θ)N[+ 1r cos(θ)B[ Z
=
−i1−κrαcos(θ)0 T[+ 1r(cos(θ)−isin(θ))N[+1r(sin(θ)+icos(θ))B[ Z
=
−i1−κrαcos(θ)0 T[+ 1re−iθN[+ 1rie−iθB[ Z.
Lemma10.5. The Laplacian of Z defined in(10.5)is given by
∆Z =
−1−κrκcos(θ)e−iθr − (1−κrα02+cos(θ))iα00 2
−iα0κ0cos(θ)+(1−κrcos(θ))α0κτsin(θ)3 r Z.
Proof. From Lemma10.4 we have
?dZ =
−i1−κrαcos(θ)0 ?T[+ 1re−iθ(?N[+i?B[) Z. Therefore,
d? dZ = Z−1dZ∧?dZ+ d
−i1−κrαcos(θ)0 ?T[+ 1re−iθ(?N[+i?B[) Z. The first part
Z−1dZ∧?dZ = −i1−κrαcos(θ)0
2 +1
re−iθ2 + 1
rie−iθ2 Z
= −(1−κrαcos(θ))02 2Z;
and for the second part we note that only d?N[ among d?T[, d?N[, d? B[ is non-zero:
d
−i1−κrαcos(θ)0 ?T[+1re−iθ(?N[+i?B[) Z
= e−iθ
r d?N[−i(1−κrαcos(θ))00 2T[∧?T[−i(1−κrαcos(θ))0 2 d(κrcosθ)∧?T[
| {z }
κ0rcosθ+κrτsinθ 1−κrcosθ ?1
− e−iθr i
−1−κrτcos(θ)T[− sin(θ)r N[+ cos(θ)r B[
∧?
N[+iB[
− er−iθ2
cos(θ)N[+sin(θ)B[
∧?
N[+iB[ Z
=
− e−iθr 1−κrκcos(θ) −i(1−κrαcos(θ))00 2 −iα0κ0cos(θ)(1−κrcos(θ))+α0κτsin(θ)3 r + er−iθ2 isin(θ)+ er−iθ2 cos(θ)− er−iθ2 cos(θ)− er−iθ2 isin(θ)
| {z }
=0
Z
The result follows when the two terms are combined.
Landau-Lifshitz Velocity
Now we are ready to evaluate ds−1(s×∆s). We continue from (10.6), which gives
s×∆s= X
X∈{T,N,B}
F
Z ×(HessF)Z (dZ(X), dZ(X))+(dF)Z(i∆Z), where we have applied (10.7). Using the formula for dZ from Lemma10.4, the Hessian term
X
X∈{T,N,B}
F
Z×(HessF)Z (dZ(X), dZ(X))
= F
Z ×(HessF)Z
−i1−κrαcos(θ)0 Z,−i1−κrαcos(θ)0 Z +F
Z ×(HessF)Z
1
re−iθZ, 1re−iθZ +F
Z
×(HessF)Z
1
rie−iθZ,1rie−iθZ .
By (10.8) the last two summand combines to zero. Now, by Lemma 10.1 the first summand becomes
= (1−κrαcos(θ))02 2(dF)Z
1−(r/r1)2 1+(r/r1)2i Z
.
Therefore, with Lemma10.5applied, the expression s×∆s becomes s×∆s = (dF)Z
−i1−κrκcos(θ)e−iθr Z+ (1−κriαcos(θ))00 2Z
+ α0κ0cos(θ)+(1−κrcos(θ))α0κτsin(θ)3 r Z −i(1−κrαcos(θ))02 2r2r+2r12Z .
Since we are interested in the value of this expression near the vortex core r = 0, let us study the first order asymptotic value for r → 0. Using the definition of Z (10.5), we have
s×∆s ∼ (dF)Z
−iκr1
1 +O(r) . Now,
ds−1(s×∆s)= (dZ−1)(dF−1)(s×∆s)
∼ dZ−1
−iκr1
1 +O(r) .
Atr =0, the pseudo-inverse dZ−1takes value in the normal plane Span{N,B}. Now, from Lemma10.4, the restriction of dZ on Span{N,B} is given by
(ιT ◦T[∧)dZ = 1
re−iθN[+ 1rie−iθB[ Z
= r11N[+r11iB[. Therefore
ds−1(s×∆s) ∼ −κB+O(r).
We conclude the calculation in this section by the following theorem.
Theorem10.2. LetΓ be a space curve in M ⊂ R3, and letU be a tubular neigh- borhood of Γ with a radius R smaller than the curvature radius of Γ. Suppose s: U → S2 wraps around S2 once conformally on each normal cross section ofU within a sufficiently small radius, then
ds−1(s×∆s) x
∼ −κB+O(dist(x,Γ)) whereκ is the curvature and B is the binormal of the curve.
The above description about s is made precise by the following construction. Let (r, ˜θ)be a polar coordinate for each normal cross section ofU withθ˜given relative to a smooth reference normal vector field ofΓ. Identify each cross section ofU with Cby Z: U → C, Z B rr1eiθ˜ with somer1 R. Then s: U → S2 is considered to be s= F◦Z whereF: C→S2is the stereographic projection.
Remark10.2. Note that the asymptotic value ds−1(s×∆s)∼ −κBis independent of the choice of the vortex thicknessr1.
Corollary 10.1. Under the assumptions in Theorem 10.2, the perturbed velocity (10.4)near the vortex core is given by
ˆ
u∼ u+2}2κB.